Is there some special case where a fermion can mediate a force?

It depends on your definition of force. Force means a change in momentum, ~dp/dt , so any change in momentum in a Feynman diagram is a force. For example this diagram for compton scattering

says yes.

If one is talking of gauge theories and exchanged bosons , because those are the ones that build up the three, electromagnetic, weak, strong ( maybe four if gravity is unified) forces , then no, by the construction.

I think this would be tricky, since any force mediator (at least from conventional thinking) must have a three-valent vertex, two of which are the charged object and one of them is the force carrier. If the force carrier is a fermion, I don't think this combination can be Lorentz invariant (spin zero combination).